User:Margav06/sandbox/Click here to continue/Time-Delay Systems/LMI for Time-Delay system on delay Independent Condition

The System
The problem is to check the stability of the following linear time-delay system



\begin{align} \begin{cases} \dot x(t)&=Ax(t)+A_dx(t-d)\\ x(t)&=\phi(t), t\in[-d,0], 0 < d \leq \bar{d},\\ \end{cases} \end{align}$$

where

\begin{align} {A, A_d}\in\mathbb{R}^{n\times n}\text{, }{A}\in\mathbb{R}^{n\times r}\text{ are the system coefficient matrices,}\\ \end{align}$$

$$ \phi (t) $$ is the initial condition $$d$$ represents the time-delay $$\bar d$$ is a known upper-bound of $$d$$

The Data
The matrices $$A, A_d$$ are known

The LMI: The Time-Delay systems (Delay Independent Condition)
From the given pieces of information, it is clear that the optimization problem only has a solution if there exists two symmetric matrices $$P,S\in\mathbb{S}^{n}$$ such that

$$P>0$$



\begin{bmatrix} A^TP+PA+S & PA_d \\ A_d^TP & -S \end{bmatrix}$$$$\begin{align}< 0\end{align}$$

This LMI has been derived from the Lyapunov function for the system. By Schur Complement we can see that the above matrix inequality is equivalent to the Riccati inequality $$ A^TP+PA+PA_dS^{-1}A_d^TP+S<0$$

Conclusion:
We can now implement these LMIs to do stability analysis for a Time delay system on the delay independent condition

Implementation
The implementation of the above LMI can be seen here

https://github.com/yashgvd/LMI_wikibooks

Related LMIs
Time Delay systems (Delay Dependent Condition)